On Characterizations of Directional Derivatives and Subdifferentials of Fuzzy Functions

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1 S S symmetry Article On Characterizations of Directional Derivatives Subdifferentials of Fuzzy Functions Wei Zhang, Yumei Xing Dong Qiu * ID College of Science, Chongqing University of Post Telecommunication, Chongqing , China; zhangwei@cqupt.edu.cn (W.Z.); x_ym2015@163.com (Y.X.) * Correspondence: dongqiumath@163.com; Tel.: ; Fax: Received: 13 July 2017; Accepted: 31 August 2017; Published: 1 September 2017 Abstract: In this paper, based on a partial order, we study the characterizations of directional derivatives the subdifferential of fuzzy function. At the same time, we also discuss the relation between the directional derivative the subdifferential. Keywords: fuzzy function; fuzzy number; directional derivative; subdifferential 1. Introduction In 1965, Zadeh [1] introduced concepts operations with respect to fuzzy sets, many authors contributed to the development of fuzzy set theory applications. Later, Zadeh proposed the fuzzy number [2 4] put forward the theory of the fuzzy numerical function together with Chang [5]. These theories those associated with optimization theory have been extensively studied in some fields, such as economics, engineering, the stock market, greenhouse gas emissions management science [6 14]. In order to solve complex optimization problems in real life, various optimization algorithms have been presented in [9,10,15 23]. Jia et al. [9] presented a new algorithm for solving the optimization problem based on the stock exchange. Afterwards, in [10,15,16], a multiple genetic algorithm multi-objective differential evolution were used to solve multiple optimization problems efficiently. Moreover, Wah et al. [17] applied a genetic algorithm to optimize flow rectification efficiency of the diffuser element based on the valveless diaphragm micropump application. Precup et al. [18] applied the grey wolf optimizer algorithm to deal with the fuzzy optimization problem. In [19], in order to solve the meta-heuristics optimization problem quickly, a bio-inspired optimization algorithm based on fuzzy logic was proposed. Peraza et al. [20] presented a new algorithm that can solve the complex optimization problems based on uncertainty management. They [22] introduced a fuzzy harmony search algorithm with fuzzy logic this algorithm was utilized to solve the fuzzy optimization problem. Amador et al. [23], presented a new optimization algorithm based on the fuzzy logic system. It is well known that convexity plays a key role in fuzzy optimization theory. Therefore, the properties of convexity of fuzzy function related problems are attached a wide range of research [24 30]. Subdifferentials are very important tools in convex fuzzy optimization theory. Based on a variety of different backgrounds, the derivative differential of fuzzy function have been widely discussed. Goetschel et al. [31,32] defined the derivative of fuzzy function, which is a generalized derivative of the set-valued function. Afterwards, Buckley et al. [33,34] defined the derivatives of fuzzy function using left- right-h functions of its α-level sets established sufficient conditions for the existence of fuzzy derivatives. Subsequently, Wang et al. [29] proposed the new concepts of directional derivative, differential subdifferential of fuzzy function from R m to E 1, discussed the characterizations of directional derivative differential of fuzzy function by using the directional derivative the differential of two crisp functions that are determined. Symmetry 2017, 9, 177; doi: /sym

2 Symmetry 2017, 9, of 12 In this paper, we investigate several characterizations of directional derivative of fuzzy function about the direction d, based on a partial order introduce the concept of the subdifferential of fuzzy function. 2. Preliminaries We denote by K C the family of all bounded closed intervals in R, that is, K C = {[a L, a R ] a L, a R R a L a R }. Given two intervals A = [a L, a R ] B = [b L, b R ], the distance between A B is defined by H(A, B) = max{ a L b L, a R b R }. Then, (K C, H) is a complete metric space [35]. Definition 1. In reference [27], suppose that E 1 = {u u : R [0, 1]} satisfies the following conditions: (1) u is normal, that is, there exists x 0 R such that u(x 0 ) = 1; (2) u is upper semicontinuous; (3) u is convex, that is, u(λx + (1 λ)y) min{u(x), u(y)} for all x, y R, λ [0, 1]; (4) [u] 0 = {x R u(x) > 0} is compact, where A denotes the closure of A. Any u E 1, is called a fuzzy number. The α-level set of fuzzy number u is a closed bounded interval [u L (α), u R (α)], where u L (α) denotes the left-h end point of [u] α u R (α) denotes the right-h end point of [u] α [36]. For u, v E 1, k R, the addition scalar multiplication are defined by, (u + v)(x) = sup min{u(s), v(t)}, s+t=x (ku)(x) = { u(k 1 x), k = 0, 0, k = 0. It is well known that for u, v E 1, k R, then u + v, ku E 1, [u + v] α = [u] α + [v] α [ku] α = k[u] α. For x = (x 1, x 2,..., x m ), y = (y 1, y 2,..., y m ) R m, it is said that x y if only if x i y i (i = 1, 2,..., m). It is said that x > y if only if x y x = y. Definition 2. In reference [29], for u, v E 1, then, (1) u v if only if [u] α = [u L (α), u R (α)] [v] α = [u L (α), u R (α)] for each α [0, 1], where [u] α [v] α if only if u L (α) v L (α) u R (α) v R (α). (2) u v if only if u v there exists α 0 [0, 1] such that u L (α 0 ) < v L (α 0 ) or u R (α 0 ) < v R (α 0 ). (3) if either u v or v u, then u v are comparable. Otherwise, u v are non-comparable. Definition 3. In reference [37], if any u, v E 1, there exists w E 1 such that u = v + w, then the stard Hukuhara difference (H-difference) of u v is defined by u v = w. Definition 4. In reference [37], (Fuzzy function) let D be a convex set of R F : D E 1 be a fuzzy function. The α-level set of F at x D, which is a closed bounded interval, can be denoted by [F(x)] α = [F L (x, α), F R (x, α)]. Thus, F can be understood by the two functions F L (x, α) F R (x, α), which are functions

3 Symmetry 2017, 9, of 12 from D [0, 1] to the set of real numbers R, F L (x, α) is a bounded increasing function of α F R (x, α) is a bounded decreasing function of α. Moreover, F L (x, α) F R (x, α) for each α [0, 1]. Definition 5. [35] For u, v E 1, the d -distance is defined by the Hausdorff metric as, d (u, v) = sup H([u] α, [v] α ) α [0,1] = sup α [0,1] max{ u L (α) v L (α), u R (α) v R (α) }. Definition 6. Let X = (a, b) let F : X E 1 be a fuzzy function {F n (x)} : X E 1, n N be a sequence of fuzzy function. If, for any ε > 0, there exists a positive integer M = M(ε) N such that, D(F n (x), F(x)) < ε for any x X, all n M. Then the sequence {F n (x)} is convergent to F(x). Definition 7. In reference [34], let F : X E 1 be a fuzzy function. Assume that the partial derivatives of F L (x, α), F R (x, α) with respect to x R for each α [0, 1] exist. The partial derivatives of F L (x, α) F R (x, α) are denoted by F L (x, α) F R (x, α), respectively. Let Γ(x, α) = [F L (x, α), F R (x, α)] for x R, α [0, 1]. Γ(x, α) defines the α-level set of fuzzy interval for x R. Then F is S-differentiable is written as, for x R, α [0, 1]. 3. Directional Derivative of the Fuzzy Function [ df(x) dx ]α = Γ(x, α) = [F L (x, α), F R (x, α)] Inspired by [29], we discuss some relations among the gradient directional derivative of fuzzy function. Moreover, several characterizations of the directional derivative of fuzzy function about the direction d are investigated, based on the partial order. Definition 8. In reference [36], (Gradient of a fuzzy function) let D be a convex set of R m F : D E 1 be a fuzzy function. For x D x (i = 1, 2,..., i m) st for the partial differentiation with respect to the ith variable x i. For each α [0, 1], F L (x, α) F R (x, α) have continuous partial derivatives so that F L(x,α) F R (x,α) x i are continuous about x. Define [ F(x) x i ] α = [ F L(x, α) x i, F R(x, α) x i ] (1) for each i = 1, 2,..., m, α [0, 1]. If for each i = 1, 2,..., m, (1) defines the α-level set of fuzzy number, then F is S-differential at x. The gradient of the fuzzy function F(x) at x, denoted by F (x), is defined as: F (x) = ( F (x), x 1 F (x),..., x 2 ) F (x). x m Remark 1. For the gradient of fuzzy function, we use the symbol, whereas for the gradient of a real valued function, we use the symbol. Definition 9. In reference [38], let D be a convex set of R m F : D E 1 be a fuzzy function. x i (1) F is convex on D if F(λx 1 + (1 λ)x 2 ) λf(x 1 ) + (1 λ)f(x 2 ) (2)

4 Symmetry 2017, 9, of 12 for any x 1, x 2 D each λ [0, 1]. (2) F is strictly convex on D if for any x 1, x 2 D with x 1 = x 2 each λ (0, 1). F(λx 1 + (1 λ)x 2 ) λf(x 1 ) + (1 λ)f(x 2 ) (3) Theorem 1. In reference [36], let D be a convex set of R m F : D E 1 be a fuzzy function, F is convex on D, if only if for each α [0, 1], F L (x, α) F R (x, α) are convex on D, that is, for each λ [0, 1], x 1, x 2 D, each α [0, 1], F L ((λx 1 + (1 λ)x 2 ), α) λf L (x 1, α) + (1 λ)f L (x 2, α) (4) F R ((λx 1 + (1 λ)x 2 ), α) λf R (x 1, α) + (1 λ)f R (x 2, α). (5) Theorem 2. Let D be a convex set of R m F : D E 1 be a S-differentiable fuzzy function. Then F is a convex fuzzy function on D if only if, for any x 1, x 2 D with (x 1 > x 2 ) such that F(x 2 ) T (x 1 x 2 ) F (x 1 ) F (x 2 ). (6) Proof. F is a convex fuzzy function on D. According to Definition 9, we obtain that: F(λx 1 + (1 λ)x 2 ) λf(x 1 ) + (1 λ)f(x 2 ) (7) for any x 1, x 2 D with x 1 > x 2, each λ [0, 1]. By Theorem 1, for each λ [0, 1] we have that F L ((λx 1 + (1 λ)x 2 ), α) λf L (x 1, α) + (1 λ)f L (x 2, α) (8) F R ((λx 1 + (1 λ)x 2 ), α) λf R (x 1, α) + (1 λ)f R (x 2, α). (9) Now combining (8) (9) imply that F L (x 2 + λ (x 1 x 2 ), α) F L (x 2, α) λ F R (x 2 + λ (x 1 x 2 ), α) F R (x 2, α) λ Taking limits for λ 0 +, we get F L (x 1, α) F L (x 2, α) (10) F R (x 1, α) F R (x 2, α). (11) F L (x 2, α) T (x 1 x 2 ) F L (x 1, α) F L (x 2, α) F R (x 2, α) T (x 1 x 2 ) F R (x 1, α) F R (x 2, α). F(x 2 ) T (x 1 x 2 ) F (x 1 ) F (x 2 ). (12) Conversely, since F is a S-differentiable fuzzy function, there exist x, y D with x > y such that F(y) T (x y) F (x) F (y).

5 Symmetry 2017, 9, of 12 For any x 1, x 2 D each λ [0, 1]. Suppose that x = x 1 y = (1 λ)x 1 + λx 2. It follows that F(y) T (x 1 y) F (x 1 ) F (y). (13) F L (y, α) T (x 1 y) F L (x 1, α) F L (y, α) (14) F R (y, α) T (x 1 y) F R (x 1, α) F R (y, α). (15) Let x = x 2 y = (1 λ)x 1 + λx 2, we get F(y) T (x 2 y) F (x 2 ) F (y). (16) F L (y, α) T (x 2 y) F L (x 2, α) F L (y, α) (17) F R (y, α) T (x 2 y) F R (x 2, α) F R (y, α). (18) Now combining (14) (1 λ) (17) λ, we have Similarly, F L (y, α) T ((1 λ)x 1 + λx 2 y) (1 λ)f L (x 1, α) + λf L (x 2, α) F L (y, α). (19) F R (y, α) T ((1 λ)x 1 + λx 2 y) (1 λ)f R (x 1, α) + λf R (x 2, α) F R (y, α). (20) The equations (19) (20) imply Therefore, F is a convex fuzzy function on D. F(λx 1 + (1 λ)x 2 ) λf(x 1 ) + (1 λ)f(x 2 ). Theorem 3. Let D be a convex set of R m F : D E 1 be a S-differentiable fuzzy function. Then F is a strictly convex fuzzy function on D if only if, for any x 1, x 2 D with (x 1 > x 2 ) such that Proof. The proof is similar to the proof of Theorem 2. F(x 2 ) T (x 1 x 2 ) F (x 1 ) F (x 2 ). (21) Theorem 4. In reference [39], let D be a convex set of R m f : D (, + ] be a convex real valued function. For x D, let d R m such that x + λd D for any λ > 0 sufficiently small. If h(λ) : (0, + ) (, + ] is defined by h(λ) = f (x + λd) f (x), λ then h(λ) is a nondecreasing function. Moreover, if f is differential at x, then lim h(λ) = lim 0 + f (x + λd) f (x) λ = f (x) T d. Definition 10. In reference [36], (directional derivative of a fuzzy function) Let D be a convex set of R m F : D E 1 be a fuzzy function. For x D, let d R m such that x + λd for any λ > 0 sufficiently small.

6 Symmetry 2017, 9, of 12 The directional derivative of F at x along the vector d (if it exists) is a fuzzy number denoted by F (x, d) whose α-level set is defined as: [ ] F α ] (x, d) = [F L ((x, d), α), F R ((x, d), α), where F F L ((x, d), α) = lim L (x + λd, α) F L (x, α) F F R ((x, d), α) = lim R (x + λd, α) F R (x, α). Theorem 5. Let D be a convex set of R m F : D E 1 be a convex S-differentiable fuzzy function. For x D, let d R m, d = (d 1, d 2,..., d m ), d i > 0, i = 1, 2,..., m, for m N. The directional derivative of F at x along the vector d is a fuzzy number denoted by F (x, d). Then F (x, d) F(x + d) F(x). Proof. Since F : D E 1 is a convex S-differentiable fuzzy function. From Theorem 2, for x D, d R m, we obtain F(x) T d F(x + d) F(x). (22) F L (x, α) T d F L (x + d, α) F L (x, α) (23) F R (x, α) T d F R (x + d, α) F R (x, α). (24) Since F is a convex fuzzy function. From Definition 10 Theorem 4, we conclude that F L F ((x, d), α) = lim L (x + λd, α) F L (x, α) = F L (x, α) T d (25) F R F ((x, d), α) = lim R (x + λd, α) F R (x, α) = F R (x, α) T d (26) Now combining (23), (24), (25) (26), we have F L ((x, d), α) = F L(x, α) T d F L (x + d, α) F L (x, α) (27) F R ((x, d), α) = F R(x, α) T d F R (x + d, α) F R (x, α). (28) The Equations (27) (28) imply F (x, d) F(x + d) F(x). Theorem 6. Let D be a convex set of R m F : D E 1 be a S-differentiable fuzzy function. For x D, let d R m such that x + λd D for any λ > 0 sufficiently small. The directional derivative of F at x along the vector d is a fuzzy number denoted by F (x, d). Then F (x, d) is a strictly positive homogeneous fuzzy function. Proof. Since the directional derivative of F at x along the vector d is a fuzzy number denoted by F (x, d). By Definition 10, we have, F F L ((x, d), α) = lim L (x + λd, α) F L (x, α)

7 Symmetry 2017, 9, of 12 For any t > 0 let τ = tλ, we obtain F F R ((x, d), α) = lim R (x + λd, α) F R (x, α). (29) F L ((x, td), α) = F lim L (x+λtd,α) F L (x,α) = [F lim L (x+τd,α) F L (x,α)]t τ 0 + τ = tf L ((x, d), α) Now combining (29) (30), we have that (30) F R ((x, td), α) = F lim R (x+λtd,α) F R (x,α) = [F lim R (x+τd,α) F R (x,α)]t τ 0 + τ = tf R ((x, d), α). [ F α (x, td)] = [F L ((x, td), α), F R ((x, td), α)] = t[f L ((x, d), α), F R ((x, d), α)] = t[f (x, d] α. Therefore, F (x, d) is a strictly positive homogeneous fuzzy function. Theorem 7. Let D be a convex set of R m F : D E 1 be a convex S-differential fuzzy function. For x D, let d R m such that x + λd D for any λ > 0 sufficiently small. The directional derivative of F at x along the vector d is a fuzzy number denoted by F (x, d). Then F (x, d) is a convex fuzzy function about the direction d. Proof. For any λ 1, λ 2 (0, 1), any d 1, d 2 R m, let λ 1 = 1 λ 2. By Definition 10 Theorem 1, we get that F L ((x, λ 1d 1 + λ 2 d 2 F ), α) = lim L(x+λ(λ 1 d1 +λ 2 d2 ),α) F L (x,α) F = lim L(λ 1 (x+λd1 )+λ 2 (x+λd2 ),α) F L (x,α) λ lim 1 [F L(x+λd1,α) F L (x,α)] + lim = λ 1 F (( L x, d 1 ), α ) + λ 2 F (( L x, d 2 ), α ) λ 2 [F L(x+λd2,α) F L (x,α)] (31) F R ((x, λ 1d 1 + λ 2 d 2 F ), α) = lim R(x+λ(λ 1 d1 +λ 2 d2 ),α) F R (x,α) F = lim R(λ 1 (x+λd1 )+λ 2 (x+λd2 ),α) F R (x,α) λ lim 1 [F R(x+λd1,α) F R (x,α)] + lim = λ 1 F (( R x, d 1 ), α ) + λ 2 F (( R x, d 2 ), α ). λ 2 [F R(x+λd2,α) F R (x,α)] (32) Hence, by Theorem 1, F (x, d) is a convex fuzzy function about the direction d. Theorem 8. Let D be a convex set of R m F : D E 1 be a convex S-differential fuzzy function. For x D, let d R m such that x + λd D for any λ > 0 sufficiently small. The directional derivative of F at x along the vector d is a fuzzy number denoted by F (x, d). Then F (x, d) is subadditive about the direction d.

8 Symmetry 2017, 9, of 12 Proof. F is a S-differential fuzzy function on D, the directional derivative of F at x along the vector d is a fuzzy number denoted by F (x, d). For arbitrary d 1, d 2 R m. By Theorem 7, we have that By Theorem 6, we obtain that F (x, 1 2 d d2 ) 1 2 F (x, d 1 ) F (x, d 2 ). (33) Now combining (33) (34), it follows that F (x, 1 2 d d2 ) = 1 2 F (x, d 1 + d 2 ). (34) F (x, d 1 + d 2 ) F (x, d 1 ) + F (x, d 2 ). Therefore, F (x, d) is subadditive about the direction d. Definition 11. In reference [40], let D be a convex set of R m F : D E 1 be a fuzzy function. Let x D, if there exists δ 0 > 0 no x U( x, δ 0 ) D such that F(x) F( x), then x is a local minimum solution of F(x). Theorem 9. Let D be a convex set of R m F : D E 1 be a fuzzy function. For x D, let d R m such that x + λd D for any λ > 0 sufficiently small. The directional derivative of F at x along the vector d is a fuzzy number denoted by F ( x, d) if 0 inter [F ( x, d)] 0, where inter A denotes the interior of the set A. Then, x is a local minimum solution of F(x). Proof. Suppose that x is not a local minimum solution of F(x). Hence, there exists a sequence {x n } n=1 any δ > 0 such that x n = x + λd U( x, δ) D ( λd < δ) F( x + λd) = F(x n ) F( x) for all n N. F L ( x + λd, α) F L ( x, α) F R ( x + λd, α) F R ( x, α). for each α [0, 1]. From Definition 10, we conclude that F L (( x, d), α) = F lim L ( x+λd,α) F L ( x,α) 0 F R (( x, d), α) = F lim R ( x+λd,α) F R ( x,α) 0. (35) (36) for each α [0, 1]. Hence, we have 0 / inter [F ( x, d)] 0. This is a contradiction with the hypothesis. Then x is a local minimum solution of F(x). 4. Subdifferential of Fuzzy Function Definition 12. (Subdifferential of a fuzzy function) Let D be a convex set of R m F : D E 1 be a fuzzy number-valued function. For x D, let d R m such that x + λd D for any λ > 0 sufficiently small. The directional derivative of F at x along the vector d is a fuzzy number denoted by F (x, d).

9 Symmetry 2017, 9, of 12 (1) A fuzzy function l(d) : R m E 1 with l(d) F ( x, d) f or all d R m. Then l(d) is called a subgradient of F at x. (2) Define the set F( x) = {l(d) l(d) F ( x, d) f or all d R m } The set F( x) of all subgradients of F at x is called the subdifferential of F at x. Now, we present some basic properties of subdifferential of fuzzy function. Theorem 10. Let D be a convex set of R m F : D E 1 be a S-differential fuzzy function. For x D, let d R m such that x + λd D for any λ > 0 is sufficiently small. The directional derivative of F at x along the vector d is a fuzzy number denoted by F (x, d). Then, F( x) is convex. Proof. Take any l 1 (d), l 2 (d) F( x) λ [0, 1]. By Definition 12, it follows that l 1 (d) F ( x, d) (37) that is, l 2 (d) F ( x, d) (37) ll 1(d, α) F L (( x, d), α) (39) lr 1 (d, α) F R (( x, d), α), (40) ll 2(d, α) F L (( x, d), α) (41) Now combining λ (39) (1 λ) (41), we have that Similarly, we obtain lr 2 (d, α) F R (( x, d), α). (42) λll 1(d, α) + (1 λ)l2 L (d, α) λf L (( x, d), α) + (1 λ)f L (( x, d), α) = F L (( x, d), α). (43) λlr 1 (d, α) + (1 λ)l2 R (d, α) λf R (( x, d), α) + (1 λ)f R (( x, d), α) = F R (( x, d), α). (44) The Equations (43) (44) imply By Definition 12, we obtain Then F( x) is convex. λ l 1 (d) + (1 λ)l 2 (d) F ( x, d). λ l 1 (d) + (1 λ)l 2 (d) F( x). Theorem 11. Let D be a convex set of R m F : D E 1 be a fuzzy function. For x D, let d R m such that x + λd D for any λ > 0 sufficiently small. The directional derivative of F at x along the vector d is a fuzzy number denoted by F (x, d). Then F( x) is closed.

10 Symmetry 2017, 9, of 12 Proof. Take an arbitrary sequence of fuzzy functions {l n (d)} n N of subgradient is convergent to fuzzy functions l(d). By Definition 6, for any ε > 0, there exists a positive integer M = M(ε) N such that for any d R m, all n M. Hence, by Definition 5, we obtain that D(l n (x), l(x)) < ε (45) sup max{ l n.l (d, α) l L (d, α), l n.r (d, α) l R (d, α) } < ε. α [0,1] Therefore, for any ε > 0, there exists a positive integer M = M(ε) N such that l n.l (d, α) l L (d, α) < ε l n.r (d, α) l R (d, α) < ε for any d R m, all n M, each α [0, 1]. lim l n.l(d, α) = l L (d, α) (46) n In view of Definition 12, we get lim l n.r(d, α) = l R (d, α) (47) n l n (d) F ( x, d). l n.l (d, α) F L (( x, d), α) (48) l n.r (d, α) F R (( x, d), α). (49) Now combining (46), (47), (48) (49), we obtain l L (d, α) F L (( x, d), α) l R (d, α) F R (( x, d), α). l(d) F ( x, d). Thus, l(d) is a subgradient. the subdifferential F( x) is closed. 5. Conclusions We have investigated several characterizations of directional derivative of fuzzy function about the direction, based on the partial order. For example, we present strictly positive homogeneity, convexity subadditivity of directional derivative of fuzzy functions. And we also propose the sufficient optimality condition for fuzzy optimization problems. Afterwards, we introduce the concept of the subdifferential of convex fuzzy function. And we present some basic characterizations of subdifferential of fuzzy function application in the convex fuzzy programming. Thus, we will apply the subdifferentiablity of fuzzy function to deal with the multiobjective fuzzy optimization problem in the future. Constrained optimization problems involving fuzzy functions are an interesting field for future study. For example, finance represents a good field to implement models for sensitive analysis through fuzzy mathematics. Several authors are working hard to shape sources of uncertainty: prices, interest rates, volatilities, etc. (see Guerra et al. [41], Buckley [42]). Therefore, fuzzy optimization problems based on parameter uncertainty sources are a topic of interest in many applications. Inspired by [20,22], directional derivatives subdifferentials of fuzzy functions will be extensively applied in some fields, such as economics, engineering, stock market greenhouse gas emission interest rates.

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